An electronically driven loudspeaker is placed near the open end of a resonance column apparatus. The length of air column in the tube is 80 cm. The frequency of the loudspeaker can be varied between 20 Hz and 2 kHz. Find the frequencies at which the column will resonate. Speed of sound in air `= 320 m s^-1`

Explanation

To find the resonant frequencies of the air column in the resonance column apparatus, we can follow these steps:

### Step 1: Identify the parameters
- Length of the air column (L) = 80 cm = 0.8 m
- Speed of sound in air (V) = 320 m/s

### Step 2: Use the formula for the fundamental frequency of a closed organ pipe
The fundamental frequency (ν₀) for a closed organ pipe is given by the formula:
\[ ν₀ = \frac{V}{4L} \]

### Step 3: Calculate the fundamental frequency
Substituting the values into the formula:
\[ ν₀ = \frac{320 \, \text{m/s}}{4 \times 0.8 \, \text{m}} = \frac{320}{3.2} = 100 \, \text{Hz} \]

### Step 4: Determine the resonant frequencies
The resonant frequencies (ν) for a closed organ pipe can be calculated using the formula:
\[ ν = (2n + 1)ν₀ \]
where \(n\) is a non-negative integer (0, 1, 2, ...).

### Step 5: Calculate the resonant frequencies for different values of n
1. For \(n = 0\):
\[ ν_0 = (2 \times 0 + 1) \times 100 = 100 \, \text{Hz} \]
2. For \(n = 1\):
\[ ν_1 = (2 \times 1 + 1) \times 100 = 300 \, \text{Hz} \]
3. For \(n = 2\):
\[ ν_2 = (2 \times 2 + 1) \times 100 = 500 \, \text{Hz} \]
4. For \(n = 3\):
\[ ν_3 = (2 \times 3 + 1) \times 100 = 700 \, \text{Hz} \]
5. For \(n = 4\):
\[ ν_4 = (2 \times 4 + 1) \times 100 = 900 \, \text{Hz} \]
6. For \(n = 5\):
\[ ν_5 = (2 \times 5 + 1) \times 100 = 1100 \, \text{Hz} \]
7. For \(n = 6\):
\[ ν_6 = (2 \times 6 + 1) \times 100 = 1300 \, \text{Hz} \]
8. For \(n = 7\):
\[ ν_7 = (2 \times 7 + 1) \times 100 = 1500 \, \text{Hz} \]
9. For \(n = 8\):
\[ ν_8 = (2 \times 8 + 1) \times 100 = 1700 \, \text{Hz} \]
10. For \(n = 9\):
\[ ν_9 = (2 \times 9 + 1) \times 100 = 1900 \, \text{Hz} \]
11. For \(n = 10\):
\[ ν_{10} = (2 \times 10 + 1) \times 100 = 2100 \, \text{Hz} \quad (\text{not valid, exceeds 2 kHz}) \]

### Step 6: List the valid resonant frequencies
The valid resonant frequencies within the range of 20 Hz to 2000 Hz are:
- 100 Hz
- 300 Hz
- 500 Hz
- 700 Hz
- 900 Hz
- 1100 Hz
- 1300 Hz
- 1500 Hz
- 1700 Hz
- 1900 Hz

### Final Answer:
The frequencies at which the column will resonate are:
- 100 Hz, 300 Hz, 500 Hz, 700 Hz, 900 Hz, 1100 Hz, 1300 Hz, 1500 Hz, 1700 Hz, and 1900 Hz.